There was very little to shout about for either side inside the opening 45 minutes, with half-chances from Wayne Rooney, Cuco Martina and Laurent Depoitre respectively but failed to kick-start the clash at Goodison Park.
The Toffees managed to take the lead just two minutes into the second period with Sigurðsson's effort, Calvert-Lewin doubled their lead in the 73rd minute to round-off what was an excellent curtain raiser for 'Big Sam'.
Few faint opportunities here and there
It had certainly been quite the week on Merseyside with Everton finally naming their successor to Ronald Koeman which was accompanied by a crucial midweek win over West Ham, this was the first game with Allardyce on the sideline and early signs looked positive but that failed to muster an opportunity of any real recognition.
Following his excellent hat-trick midweek Rooney looked to make the difference once again in the 20th minute, the ball broke which allowed the striker to shoot which was blocked by Christopher Schindler. The chance was still on as it looped in the air, Rooney looked to challenge Scott Malone in the air but he leads with his elbow and is blown up by Chris Kavanagh.
Martina looked to add to his only league goal thus far in the 25th minute as he played a neat one-two with Dominic Calvert-Lewin, he got his shot off but it was too easy for Jonas Lossl as he palmed it away.
The best chance of the half actually fell to the visitors who were looking to reignite their form which has escaped them in recent clashes, Tom Ince's low ball was into the feet of Depoitre but Jordan Pickford was quick to smother the effort before Ashley Williams cleared.
Making the best possible to the restart
Allardyce's first half-time team talk will have still been ringing in the ears of the home players as they came out for the second period, and it certainly seemed to work as they took the lead inside two minutes.
It was swift play as Aaron Lennon ran inside before playing it off to Calvert-Lewin, the youngster's excellent flick was into the path of the onrushing Sigurðsson and was as cold as ice as he stroked it into the bottom corner.
Falling behind looked to have kicked David Wagner's side into gear as they broke quickly in the 51st minute, they broke through Ince down the left but Martina did well to hold him up and was enough as his effort was into the side-netting.
Doubling it up
Everton were looking comfortable throughout the clash as they looked to make sure the three points were secured, and did just that with their second in the 73rd minute.
Idrissa Gueye did well to win the ball in the midfield as he fed it into Rooney, his pass is brilliantly threaded into Calvert-Lewin who steadied himself before a deflection from Zanka looped it over Lossl and into the net.
Much like after the first Huddersfield looked to give an instant reply, as Aaron Mooy's corner evaded Kasey Palmer but fell for Tommy Smith but the long-range effort was held onto by Pickford.